Pick and Freeze estimation of sensitivity index for static and dynamic models with dependent inputs

Grandjacques, Mathilde; Delinchant, Benoit; Adrot, Olivier

Journal de la société française de statistique, Tome 157 (2016) no. 2, pp. 65-89.

Cet article traite de l’estimation des indices de Sobol pour des entrées dépendantes, statiques et dynamiques. Nous proposons de transformer l’entrée dépendante en une image dont les composantes sont indépendantes. Elles ont la propriété de vérifier l’invariance des espérances conditionnelles sachant le sous espace formé des entrées et celui de leurs images, ce qui nous permet d’appliquer la méthode Pick and Freeze au modèle.

Nous traitons tout d’abord l’aspect statique, en détaillant le cas général non Gaussien et le cas Gaussien. Dans le cas non Gaussien, nous appliquons la méthode dite des quantiles conditionnels, généralement utilisée pour simuler des vecteurs aléatoires. Dans le cas Gaussien, les variables dépendantes sont séparées en deux groupes de variables indépendantes.

Concernant l’aspect dynamique, la définition des indices a été légèrement modifiée afin de prendre en compte les deux dimensions de dépendance de l’entrée, la dimension temporelle et la dimension spatiale. Pour les processus Gaussiens, nous utilisons la même méthode que dans le cas statique. Pour les processus non Gaussiens, nous proposons d’utiliser un modèle à base copule pour revenir à des entrées Gaussiennes.

L’étude de différents cas met en évidence le fait que, dans les études de sensibilité, l’utilisation de la corrélation comme mesure de dépendance a ses limites.

This article addresses the estimation of the Sobol index for dependent static and dynamic inputs. We study transformations in the input, whose image is an input with independent components. They have the basic property to give an invariance property for conditional expectation between a subset of inputs and their image that allows to use the Pick and Freeze method.

We first focus on the static case. The general case and the Gaussian case are detailed . In the non Gaussian case we apply the conditional quantile function generally used to simulate random vectors in a new framework. In the Gaussian case the dependent variables are separated into two groups of independent variables.

In the dynamic case the definition of the index has been slightly modified in order to take into account the two dimensions of dependence (temporal and spatial). For Gaussian processes the same method as previously is used. For non Gaussian processes for which in general there is no sufficient information to get a model, we propose to use a copula model to get back to Gaussian inputs. Different cases are studied in order to underline on the weakness, in sensitivity studies, to use the correlations like the measures of dependence.

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